| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.8 This is a Further Maths statistics question requiring confidence intervals for the difference of two means with known population variance. While the calculations are standard (parts a-b), part (c) requires working backwards from a hypothesis test condition to find a critical confidence level, which demands deeper understanding of the relationship between confidence intervals and significance levels. The multi-step nature and the reverse-engineering in part (c) elevate this above routine A-level questions. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (SE of difference of means) \(= \sqrt{\frac{40^2}{12} + \frac{40^2}{10}}\) | M1 | Award M1 for \(\text{Var} = \frac{40^2}{12} + \frac{40^2}{10}\) |
| \(= 17.1(2697677)\) | A1 | — |
| 98% CI | — | — |
| \(30 \pm 2.3263\sqrt{\frac{40^2}{12} + \frac{40^2}{10}}\) | M1 | Or 2.326 from tables |
| \([-9.84, 69.84]\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| We cannot conclude that either protein powder is better than the other in promoting weight gain. Because the confidence interval contains 0 | E1 E1 | FT their CI |
| Answer | Marks | Guidance |
|---|---|---|
| \(30 - k\sqrt{\frac{40^2}{12} + \frac{40^2}{10}} > 0\) | M1 | Condone \(=\) |
| \(k < 1.7516...\) | A1 | FT their SE from (a) and their difference in means for possible M1A1A1A1 |
| Probability from calculator \(= 0.96008\) Or \(0.95994\) from tables | A1 | — |
| Confidence level 92% | A1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| Valid assumption e.g. Rest of the diet is the same. They exercise the same amount. They follow the same program for muscle gain. | E1 | — |
## (a)
(SE of difference of means) $= \sqrt{\frac{40^2}{12} + \frac{40^2}{10}}$ | M1 | Award M1 for $\text{Var} = \frac{40^2}{12} + \frac{40^2}{10}$
$= 17.1(2697677)$ | A1 | —
98% CI | — | —
$30 \pm 2.3263\sqrt{\frac{40^2}{12} + \frac{40^2}{10}}$ | M1 | Or 2.326 from tables
$[-9.84, 69.84]$ | A1 | cao
## (b)
We cannot conclude that either protein powder is better than the other in promoting weight gain. Because the confidence interval contains 0 | E1 E1 | FT their CI
## (c)
$30 - k\sqrt{\frac{40^2}{12} + \frac{40^2}{10}} > 0$ | M1 | Condone $=$
$k < 1.7516...$ | A1 | FT their SE from (a) and their difference in means for possible M1A1A1A1
Probability from calculator $= 0.96008$ Or $0.95994$ from tables | A1 | —
Confidence level 92% | A1 | —
## (d)
Valid assumption e.g. Rest of the diet is the same. They exercise the same amount. They follow the same program for muscle gain. | E1 | —
**Total: [11]**
---Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players.
Protein powder $A$ was used by 12 players, randomly selected, and their mean weight gain was 900 g. Protein powder $B$ was used by the other 10 players and their mean weight gain was 870 g.
Let $\mu_A$ and $\mu_B$ be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder $A$ and protein powder $B$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Calculate a 98\% confidence interval for $\mu_A - \mu_B$. [4]
\item In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer. [2]
\item Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder $A$ over protein powder $B$. [4]
\item State one non-statistical assumption you have made in order to reach these conclusions. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2019 Q4 [11]}}